2005 Cyprus Seniors TST/Day 1/Problem 3
Problem
Given a circle with ceneter 
and an inscribed trapezium 
with 
. If 
is a point of arc 
on which 
and 
do not belong to, and 
and 
are the projections of 
on the lines 
, 
, 
, and 
respectively, show that:
(i) The circumscribed circles of the triangles 
and 
intersect on the side .
(ii) The points and are concyclic.
(iii) If 
, 
and the distance between the parallel chords is 
find all the points of the axis of symmetry of the trapezium that can `see'* at right angle the non parallel sides and calculate their distance form 
and in terms of 
, 
and .
Examinate if such pints always exist. \newline
(Draw a separate diagram for part (iii)).
*An example what I mean: In any cyclic quadrilateral 
the point and `see' the side 
at the same angle.
